Several theorems in the theory of automata and languages only depend on a few equational properties of fixed point operations. This has been shown, for example, for the classical Kleene theorem equating the class of regular languages with the class of recognizable languages. One objective of the proposed research is to further develop and apply the general theory of fixed points in connection with automata so as to provide an axiomatic foundation to the theory of automata and languages over finite and infinite words, trees, and formal series, and various bisimulation based semantics of transition systems.

Language varieties and the Eilenberg correspondence have played an important role in the classification of regular languages. The Eilenberg correspondence establishes a bijection between language varieties and pseudo-varieties of certain finite algebraic structures. Many deep results of the theory of automata and languages are concerned with language varieties and their corresponding pseudo-varieties. Recently, variety theory has been extended to trees which are frequently used to describe the behavior of concurrent systems. Several programming logics have been introduced for the specification of tree languages. However, a satisfactory (algorithmic) characterization has been obtained only in a few cases. Since most of these logics determine tree language varieties, the characterization of their expressive power is equivalent to the characterization of the corresponding pseudo-variety. If this pseudo-variety is decidable, there results an effective characterization of the logic. We plan to study varieties of tree languages both in connection with logic and independently.

We have proved that the the bahavior of finite
automata (tree automata, weighted automata, etc.)
has a finite description with respect to the general
properties of fixed point operations. We have obtained
complete axiomatizations of rational power
series and tree series, and the bisimulation based
behavior of finite automata. As an intermediate step
of the completeness proofs, we have shown that
Kleene's fundamental theorem and its generalizations
follow from the equational properties of fixed point operations.
We have developed an algebraic framework for
describing the expressive power of branching time temporal logics
and fragments of monadic second-order logic on trees. Our
main results establish a bijective correspondence between
these logics and certain pseudo-varieties of finite algebras
and/or finitary preclones.
We have characterized the lexicographic orderings of the
regular and context-free languages and generalized the notion
of context-free languages to infinite words and established several
of their algorithmic properties.